Kramers–Moyal expansion


In stochastic processes, Kramers–Moyal expansion refers to a Taylor series expansion of the master equation, named after Hans Kramers and José Enrique Moyal. This expansion transforms the integro-differential master equation
where is the transition probability density, to an infinite order partial differential equation
where
Here is the transition probability rate. Fokker–Planck equation is obtained by keeping only the first two terms of the series in which is the drift and is the diffusion coefficient.

Pawula theorem

The Pawula theorem states that the expansion either stops after the first term or the second term. If the expansion continues past the second term it must contain an infinite number of terms, in order that the solution to the equation be interpretable as a probability density function.

Implementations